College Physics

Figure 10.4In circular motion, linear accelerationa, occurs as the magnitude of the velocity changes:ais tangent to the motion. In the context of circular motion, linear

acceleration is also called tangential accelerationat.

Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know fromUniform Circular Motion and

Gravitationthat in circular motion centripetal acceleration,ac, refers to changes in the direction of the velocity but not its magnitude. An object

undergoing circular motion experiences centripetal acceleration, as seen inFigure 10.5. Thus,atandacare perpendicular and independent of

one another. Tangential accelerationatis directly related to the angular accelerationαand is linked to an increase or decrease in the velocity, but

not its direction.

Figure 10.5Centripetal accelerationacoccurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus

perpendicular to each other.

Now we can find the exact relationship between linear accelerationatand angular accelerationα. Because linear acceleration is proportional to a

change in the magnitude of the velocity, it is defined (as it was inOne-Dimensional Kinematics) to be

a (10.10)

t=

Δv

Δt

.

For circular motion, note thatv=rω, so that

(10.11)

at=

Δ(rω)

Δt

.

The radiusris constant for circular motion, and soΔ(rω) =r(Δω). Thus,

(10.12)

at=rΔω

Δt

.

By definition,α=Δω

Δt

. Thus,

at=rα, (10.13)

or

(10.14)

α=

at

r.

These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger

the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car’s drive wheels, the greater the

acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration

α.

322 CHAPTER 10 | ROTATIONAL MOTION AND ANGULAR MOMENTUM

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